Subset sums over Galois rings II

Abstract

Suppose that p is an odd prime, and R=GR(p2,p2r) is a Galois ring of characteristic p2, whose cardinality is p2r. For any subset D⊆R, let N(D,n,b) and Nf(D,n,b), respectively, be the number of n-subsets, denoted by S, in D such that ∑x∈Sx=b and ∑x∈Sf(x)=b, where f(x)∈R[x] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N(D,n,b) and Nf(D,n,b) under some certain constraints of D and f(x). The special cases D={ak:a∈R⁎} and f(x)=xk were studied by the authors in a previous paper, where R⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings.